Covered in lectures. Check back once the chapter is concluded.
1 Geometry of the Complex plane
This section is a brief reminder of Sections 3 and 4 of MA1006 Algebra.
Definition 1.1 The complex numbers \(\C\) are the set of all pairs \(z=(x,y)\in\R^2\) of real numbers with the addition
\[ z_1+z_2=(x_1+x_2,y_1+y_2) \tag{1.1}\]
and the multiplication
\[ z_1\cdot z_2=(x_1x_2-y_1y_2,x_1y_2+x_2y_1), \tag{1.2}\]
where \(z_1=(x_1,y_1)\) and \(z_2=(x_2,y_2).\) We call \(x=\Re(z)\) the real part and \(y=\Im(z)\) the imaginary part of \(z.\)
From Section 3.8 in MA1006 Algebra we recall:
Proposition 1.1 The complex numbers are a field.
Remark 1.1.
Covered in lectures. Check back once the chapter is concluded.
Of course, \(\C\) is a one-dimensional vector space over itself. Restricting the scalar multiplication to \(\R\) makes \(\C\) into a vector space over \(\R,\) isomorphic to \(\R^2,\) of dimension two with standard basis \(1,i\in\C.\)
Proposition 1.2
In the standard basis, every \(A=\begin{pmatrix}a&b \\ c&d\end{pmatrix} \in\M_{2\t2}(\R)\) corresponds uniquely to an \(\R\)-linear map \(T_A\colon\C\to\C,\) namely \[ T_A(x+iy)=(ax+by)+i(cx+dy). \tag{1.3}\]
\(T_A\) is \(\C\)-linear \(\iff\) \(a=d\) and \(b=-c.\) In this case, \[ T_A(z)=\al\cdot z,\qquad \al=a+ic. \tag{1.4}\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Definition 1.2 The conjugate of \(z\in\C\) is the complex number \[\ol{z}=(x,-y),\]and the modulus (also called absolute value) is\[|z|=\sqrt{x^2+y^2}\geqslant0.\]
Proposition 1.3 The following formulas hold for \(z,w\in \C:\)
\[\begin{align} \ol{z\cdot w}&=\ol{z}\cdot\ol{w} &\ol{z+w}&=\ol{z}+\ol{w}\\ \ol{\ol{z}}&=z &\ol{i}=-i&,\ \ol{1}=1\\ z\cdot\ol{z}&=|z|^2 &|z\cdot w|&=|z|\cdot|w|\\ \Re(z)&=\frac{z+\ol{z}}{2} &\Im(z)&=\frac{z-\ol{z}}{2i}\\ z^{-1}&=\frac{\ol{z}}{|z|^2}\quad\text{if $z\neq0$} \end{align}\]
Covered in lectures. Check back once the chapter is concluded.
Proposition 1.4 The following inequalities hold for \(z,w\in\C:\)
\[|z+w|\leqslant|z|+|w| \qquad |z-w|\geqslant\bigl||z|-|w|\bigr| \tag{1.5}\]
\[|\Re(z)|\leqslant|z| \qquad |\Im(z)|\leqslant|z| \tag{1.6}\]
Covered in lectures. Check back once the chapter is concluded.
Proposition 1.5 For every non-zero complex number \(z=(x,y)\) there is an argument \(\th\in\R\) and a radius \(r=|z|>0\) such
\[x=r\cos(\th), y=r\sin(\th). \tag{1.7}\]
This representation is unique up to replacing \(\th\) by \(\th+2\pi k\) for any \(k\in\Z.\)
Proof. (omitted)
Since \(x^2+y^2=r^2(\cos(\th)^2+\sin(\th)^2)=r^2,\) the radius must be \(r=|z|.\) Define the complex number \(w=r^{-1}z\) and write \(w=u+iv\) for its real and imaginary parts.
We will prove the existence of \(\th\in\R\) with \(u=\cos(\th),\) \(v=\sin(\th).\) This also proves the existence of a representation Equation 1.7, by multiplying by \(r.\) Since \(u^2+v^2=|w|^2=r^{-2}|z|^2=1,\) we know \(|u|\leqslant1,\) \(|v|\leqslant1.\) Recall that \(\cos\colon[0,\pi]\to[-1,1]\) and \(\sin\colon[-\pi/2,\pi/2]\to[-1,1]\) are bijections. Hence \[\begin{align*} u&=\cos(\al) &&\text{ for some }\al\in[0,\pi], \\ v&=\sin(\be) &&\text{ for some }\be\in[-\pi/2,\pi/2]. \end{align*}\] As \(\sin(\be)^2=v^2=1-u^2=1-\cos(\al)^2=\sin(\al)^2,\) we have \(\sin(\al)=\pm\sin(\be)=\sin(\pm\be).\) To produce the correct \(\th,\) we distinguish two cases.
- Case 1
- \(\al\in[0,\pi/2].\) Then \(\al=\pm\be\) by the injectivity of the sine function on the interval \([-\pi/2,\pi/2].\) Setting \(\th=\pm\al=\be,\) we find that \(u=\cos(\th)\) and \(v=\sin(\th),\) as required.
- Case 2
- \(\al\in[\pi/2,\pi].\) Then \(\pi-\al, \be\in[-\pi/2,\pi/2]\) and \(\sin(\pi-\al)=\sin(\al)=\pm\sin(\be),\) so \(\pi-\al=\pm\be\) by injectivity. Setting \(\th=\pm\al=\pm\pi-\be,\) we find \(u=\cos(\th), v=\sin(\th),\) using trigonometric identities.
This completes the existence part of the proof. For uniqueness, we already know that \(r=|z|>0\) is unique, so it remains to consider \[\begin{align*} x&=r\cos(\th_1)=r\cos(\th_2), &y&=r\sin(\th_1)=r\sin(\th_2). \end{align*}\] To translate the situation into an interval that we understand, pick \(k_1, k_2\in \Z\) so that \(\th_1 + 2\pi k_1, \th_2 + 2\pi k_2 \in [-\pi,\pi).\) Then \[\cos(\th_1+2\pi k_1)=\cos(\th_1)=\cos(\th_2)=\cos(\th_2+2\pi k_2).\]
Using the injectivity of the cosine function and considering cases as above, we find that \(\th_1+2\pi k_1 = \pm(\th_2+2\pi k_2).\) If the sign is ‘\(+\)’ we get \(\th_1-\th_2=2\pi(k_2-k_1)\) and we are done, so suppose \(\th_1+2\pi k_1 = -(\th_2+2\pi k_2).\) Then \[\sin(\th_1)=\sin(\th_2)=\sin(\th_2+2\pi k_1)=-\sin(\th_1+2\pi k_1)=-\sin(\th_1)\]
implies \(\sin(\th_1)=0.\) Therefore \(\th_1\) is a multiple of \(2\pi,\) which implies that \(\th_1+2\pi k_1 = -(\th_2+2\pi k_2)=0\) since these numbers were chosen in \([-\pi,\pi)\) and we have \(2\pi\Z\cap[-\pi,\pi)=\{0\}.\) Hence \(\th_1-\th_2=2\pi(k_2-k_1).\)
To get around the non-uniqueness of the argument in polar coordinates, we restrict \(\th\) to lie in a half-open interval of length \(2\pi.\) Here is the most common convention.
Definition 1.3 The principal argument of a non-zero \(z\in\C\) is the unique \(\th\in(-\pi,\pi]\) such that Equation 1.7 holds, and we write \(\arg(z)=\th.\)
Definition 1.4 The value of the exponential function at the complex number \(z=x+i\th,\) where \(x,\th\in\R,\) is defined as \[e^{x+i\th}=e^x\bigl(\cos(\th)+i\sin(\th)\bigr). \tag{1.8}\]
Covered in lectures. Check back once the chapter is concluded.
Proposition 1.6 \(e^{z_1}\cdot e^{z_2}=e^{z_1+z_2}\) for all \(z_1, z_2\in\C.\) Moreover, we have \((e^z)^n=e^{nz}\) for all \(z\in\C,\) \(n\in\Z.\)
Proof.
Covered in lectures. Check back once the chapter is concluded.
The polar form can be applied to the construction of \(n\)th roots.
Covered in lectures. Check back once the chapter is concluded.
Proposition 1.7 Every complex number \(z\neq0\) has an \(n\)th root \(w\) satisfying \(w^n=z.\) If \(w\) is an \(n\)th root of \(z,\) the set of all \(n\)th roots of \(z\) is \[\bigl\{w, \ze_n\cdot w, \ze_n^2\cdot w,\ldots, \ze_n^{n-1}\cdot w\bigr\}. \]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Questions for further discussion
- The complex numbers are obtained by ‘adjoining’ a symbol \(i\) with \(i^2=-1.\) If instead we would have adjoined a different symbol \(\ep\) with \(\ep^2=-1,\) would the set of elements \(x+\ep y\) still define a field?
- The real numbers have a total order ‘\(\leqslant\)’. Why doesn’t it make sense to extend this definition to the complex numbers?
- Describe geometrically the set \(R_n=\{1,\ze_n,\ldots,(\ze_n)^{n-1}\}\) of \(n\)th roots of unity. Find a connection between \(R_n\) and the cyclic group \(C_n=\{\ol{0},\ldots,\ol{n-1}\}\) of integers modulo \(n\) from MX3020 Group Theory.
1.1 Exercises
This problem sheet is intended as a recap and contains more problems than can be discussed during the tutorials.
Verify \[z = x+iy\] and \[i^2 =-1\] straight from the definition Equation 1.2.
How many real solutions \(x\) does \(x^2+1=0\) have? Show that the polynomial equation \(z^2+1=0\) has exactly two solutions \(z\in\C.\)
Give examples of complex numbers \(z,w\neq0\) such that \(z^2+w^2=0.\)
Sketch the position of the complex numbers \(i, 1+i, \frac{3+2i}{4}\) in the plane.
Express the following complex numbers \(z\) in the form \(x+iy\) with \(x,y\in\R.\) \[(1+i)^{20},\ (5+3i)(1+2i),\ (1-i)(2+3i),\ (1-i)i(1+i),\ \frac{2+i}{1-i}\]
Express the following complex numbers \(z\) in the form \(x+iy\) with \(x,y\in\R.\) \[1/i,\ \frac{1}{1+i},\ \frac{3+i}{3-i} \]
Find the modulus and the conjugate of the following complex numbers. \[2+i,\ i,\ 5-3i,\ \frac{1+i}{2+i} \]
Describe the sets \(A=\{z\in\C\mid\Im(z)>0\},\) \(B=\{z\in\C\mid\Re(z)\leqslant1\},\) \(C=\{z\in\C\mid \Re((1+i)z)=0\},\) and \(A\cap B\) geometrically.
Describe the set \(D=\{z\in\C \mid z\cdot\overline{z}=1\}\) geometrically.
Hint: Write \(z=re^{i\th}\) in polar form.
Draw all nine sets described by the following conditions on the complex number \(z.\) \[\begin{align*} |z|&=1,&|z|&<1,&1<&|z|<2,\\ |1+z|&>1,&|2-z|&<2,&3<&|z+i|<4,\\ |z-1|&<|z+1|,&|z|&=|z+1|,&|z-1|&=|z+i|. \end{align*}\]
Let \(S=\{x+iy\in \C \mid 0\leqslant x,y \leqslant 1\}\). Draw \(S\) and the sets \[\begin{align*} A&=\{2z \mid z\in S\}, &B&=\{\ol{z} \mid z\in S\},\\ C&=\{-z \mid z\in S\}, &D&=\{z^2 \mid z\in S\}. \end{align*}\]
Let \(D=\{z\in\C \mid |z|<1\}\) be the unit disk. Draw the sets \[A=\{2z \mid z\in D\},\quad B=\{z^2 \mid z\in D\},\quad C=\{|z| \mid z\in D\}.\]
Show that \(i=e^{i\pi/2}\) and \(-1=e^{i\pi}.\)
Express the following complex numbers \(z\) in the form \(x+iy\) with \(x,y\in\R.\) \[e^{i\pi/4},\ e^{i\pi},\ e^{i\frac{2\pi}{3}} \]
Write each of the following complex numbers in polar form \(re^{i\theta}\) with \(r>0\) and \(-\pi<\theta\leqslant\pi.\) \[i, \quad -1, \quad -i, \quad 1+i, \quad 1-i, \quad i-1, \quad \frac{1}{2}+i\frac{\sqrt{3}}{2}\] Draw each of these numbers in the complex plane.
Calculate \(i^{2021}\) and \((1+i)^{20}.\)
Solve the equation \((1 - i)^n - 2075 = 2021\) and find \(n\in\N.\)
Prove that for \(z\in\R\) we have
\[\begin{align*} \cos(z)&=\frac{e^{iz}+e^{-iz}}{2}, &\sin(z)&=\frac{e^{iz}-e^{-iz}}{2i}. \end{align*}\]
Use these equations to extend the definition of the functions \(\cos(z), \sin(z)\) to complex arguments \(z\in\C.\) Find \(z\in\C\) with \(\sin(z)=2.\)
Hint: Put \(w=e^{iz}\) and reduce to a quadratic equation.
Prove the following statements.
- \(\ol{zw}=\ol{z}\cdot\ol{w}\) for all \(z,w\in\C\)
- \(\ol{z_1z_2\cdots z_n} = \ol{z}_1\ol{z}_2\cdots\ol{z}_n\) for all \(z_1, z_2, \ldots, z_n\in\C\) (use induction)
- \(\ol{(z^n)}=(\ol{z})^n\) for all \(z\in\C\)
- Let \(p(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0\) be a polynomial with real coefficients \(a_0,\ldots,a_n\in\R.\) Prove that \(\ol{p(z)} = p({\ol{z}}).\) Deduce that all roots of \(p(z)\) occur in complex conjugate pairs.
Show that \(|z|=|{-z}|\) and \(|\ol z|=|z|.\) Prove also that \(|\la z|=\la |z|\) for all \(\la\geqslant 0.\)
Prove that \(\Re(z)=\frac{1}{2}(z+\bar{z}),\) \(\Im(z)=\frac{1}{2i}(z-\bar{z}).\)
Prove that \(\ol{e^z}=e^{\ol z}.\) Deduce that \(|e^z|=e^{\Re(z)}.\)
Prove that \(|z+w|^2+|z-w|^2=2|z|^2+2|w|^2.\)
Show that \(|z+w|^2=|z|^2+2\Re(z\ol w)+|w|^2.\) Use this to determine the conditions on \(z,w\) for \(|z+w|=|z|+|w|\) to hold.
Assuming we know the triangle inequality \(|z+w|\leqslant |z|+|w|\) for all \(z,w\in\C,\) prove the reverse triangle inequality \[|z-w|\geqslant\bigl||z|-|w|\bigr|.\]
Let \(K\) be a field with \(\R\subset K \subset\C.\) Prove that \(K=\R\) or \(K=\C.\)
Further resources
- Freitag–Busam (Freitag and Busam 2009, chap. 1) for additional exercises and historical background.
- https://en.wikipedia.org/wiki/Complex_number for overview and history
- https://youtu.be/T647CGsuOVU for some visualization